Some Tauberian theorems for iterations of Hölder integrability method
Articles > Article · 2019
Authors: Zerrin Önder, İbrahim ÇANAK
Positivity, Journal Indexed in SCI Expanded, International, WoS, 2019, vol 23, pp 1179-1193
DOI: 10.1007/s11117-019-00660-w
Summary
Let f be a real or complex-valued function on [1 , ∞) which is continuous over every finite interval [1, x) for 1 < x< ∞. We set s(x):=∫1xf(t)dt and define σk(s(x)) by σk(s(x))={1x∫1xσk-1(s(t))dt,k≥1s(x),k=0for each nonnegative integer k. An improper integral ∫1∞f(x)dxis said to be integrable to a finite number μ by the k-th iteration of Hölder or Cesàro mean method of order one, or for short, the (H, k) integrable to μ if limx→∞σk(s(x))=μ.In this case, we write s(x)→μ(H,k). It is clear that the (H, k) integrability method reduces to the ordinary convergence for k= 0 and the (H, 1) integrability method is (C, 1) integrability method. It is known that lim x→∞s(x) = μ implies lim x→∞σk(s(x)) = μ. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Hölder integrability method under which the converse implication holds.